Monday, January 18, 2010

John Geanakoplos on the Leverage Cycle

In a series of papers starting with Promises Promises in 1997, John Geanakoplos has been developing general equilibrium models of asset pricing in which collateral, leverage and default play a central role. This work has attracted a fair amount of media attention since the onset of the financial crisis. While the public visibility will surely pass, I believe that the work itself is foundational, and will give rise to an important literature with implications for both theory and policy.

The latest paper in the sequence is The Leverage Cycle, to be published later this year in the NBER Macroeconomics Annual. Among the many insights contained there is the following: the price of an asset at any point in time is determined not simply by the stream of revenues it is expected to yield, but also by the manner in which wealth is distributed across individuals with varying beliefs, and the extent to which these individuals have access to leverage. As a result, a relatively modest decline in expectations about future revenues can result in a crash in asset prices because of two amplifying mechanisms: changes in the degree of equilibrium leverage, and the bankruptcy of those who hold the most optimistic beliefs.

This has some rather significant policy implications:

In the absence of intervention, leverage becomes too high in boom times, and too low in bad times. As a result, in boom times asset prices are too high, and in crisis times they are too low. This is the leverage cycle.

Leverage dramatically increased in the United States and globally from 1999 to 2006. A bank that in 2006 wanted to buy a AAA-rated mortgage security could borrow 98.4% of the purchase price, using the security as collateral, and pay only 1.6% in cash. The leverage was thus 100 to 1.6, or about 60 to 1. The average leverage in 2006 across all of the US$2.5 trillion of so-called ‘toxic’ mortgage securities was about 16 to 1, meaning that the buyers paid down only $150 billion and borrowed the other $2.35 trillion. Home buyers could get a mortgage leveraged 20 to 1, a 5% down payment. Security and house prices soared.

Today leverage has been drastically curtailed by nervous lenders wanting more collateral for every dollar loaned. Those toxic mortgage securities are now leveraged on average only about 1.2 to 1. Home buyers can now only leverage themselves 5 to 1 if they can get a government loan, and less if they need a private loan. De-leveraging is the main reason the prices of both securities and homes are still falling.

Geanakoplos concludes that the Fed should actively "manage system wide leverage, curtailing leverage in normal or ebullient times, and propping up leverage in anxious times." This seems consistent with Paul Volcker's views (as expressed in his 1978 Moskowitz lecture) and with Hyman Minsky's financial instability hypothesis. But it is inconsistent with the adoption of any monetary policy rule (such as the Taylor rule) that is responsive only to inflation and the output gap.

It is worth examining in some detail the theoretical analysis on which these conclusions rest. Start with a simple model with a single asset, two periods, and two future states in which the asset value will be either high or low. Beliefs about the relative likelihood of the two states vary across individuals. These belief differences are primitives of the model, and not based on differences in information (technically, individuals have heterogeneous priors). Suppose initially that there is no borrowing. Then the price of the asset will be such that those who wish to sell their holdings at that price collectively own precisely the amount that those who wish to buy can collectively afford. Specifically, the price will partition the public into two groups, with those more pessimistic about the future price selling to those who are more optimistic.

Now allow for borrowing, with the asset itself as collateral (as in mortgage contracts). Suppose, for the moment, that the amount of lending is constrained by the lowest possible future value of the collateral, so lenders are fully protected against loss. Even in this case, the asset price will be higher than it would be without borrowing: the most optimistic individuals will buy the asset on margin, while the remainder sell their holdings and lend money to the buyers. Already we see something interesting: despite the fact that there has been no change in beliefs about the future value of the asset, the price is higher when margin purchases can take place:

The lesson here is that the looser the collateral requirement, the higher will be the prices of assets... This has not been properly understood by economists. The conventional view is that the lower is the interest rate, then the higher will asset prices be, because their cash flows will be discounted less. But in the example I just described... fundamentals do not change, but because of a change in lending standards, asset prices rise. Clearly there is something wrong with conventional asset pricing formulas. The problem is that to compute fundamental value, one has to use probabilities. But whose probabilities?

The recent run up in asset prices has been attributed to irrational exuberance because conventional pricing formulas based on fundamental values failed to explain it. But the explanation I propose is that collateral requirements got looser and looser.

So far, the extent of leverage has been assumed to be fixed (either at zero or at the level at which the lender is certain to be repaid even in the worst-case outcome). But endogenous leverage is an important part of the story, and the extent of leverage must be determined jointly with the interest rate in the market for loans. To accomplish this, one has to recognize that loan contracts can differ independently along both dimensions:

It is not surprising that economists have had trouble modeling equilibrium haircuts or leverage. We have been taught that the only equilibrating variables are prices. It seems impossible that the demand equals supply equation for loans could determine two variables.

The key is to think of many loans, not one loan. Irving Fisher and then Ken Arrow taught us to index commodities by their location, or their time period, or by the state of nature, so that the same quality apple in different places or different periods might have different prices. So we must index each promise by its collateral...

Conceptually we must replace the notion of contracts as promises with the notion of contracts as ordered pairs of promises and collateral.

Even though the universe of possible contracts is large, only a small subset of these contracts will actually be traded in equilibrium. In the simple version of the model considered here, equilibrium leverage is uniquely determined (given the distribution of beliefs about future asset values).

To derive the amplifying mechanisms which give rise to the leverage cycle, the model must be extended to allow for three periods. In each period after the initial one the news can be good or bad, so there are now four possible paths through the tree of uncertainly. As before, suppose that at the end of the final period the asset price can be either high or low, and that it will be low only if bad news arrives in both periods. Short term borrowing (with repayment after one period) is possible, and the degree of leverage in each period is determined in equilibrium. It turns out that in the first period the equilibrium margin is just enough to protect lenders from loss even if the initial news is bad. The most optimistic individuals borrow and buy the asset, the remainder sell what they hold and lend.

Now suppose that the initial news is indeed bad. Geanakoplos shows that the asset price will fall dramatically, much more than changing expectations about its eventual value could possibly warrant. This happens for two reasons. First, the most optimistic individuals have been wiped out and can no longer afford to purchase the asset at any price. And second, the amount of equilibrium leverage itself falls sharply. There is less borrowing by less optimistic individuals resulting in a much lower price than would arise if those who had borrowed in the initial period had not lost their collateral.

There is much more in the paper than I have been able to describe, but these simple examples should suffice to illuminate some of the key ideas. As I said at the start of this post, I suspect that a lot of research over the next few years will build on these foundations. There is still a large gap between the rigorous and tightly focused analysis of Geanakoplos on the one hand, and the expansive but informal theories of Minsky on the other. An attempt to bridge this gap seems like it would be a worthwhile endeavor.

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Update (1/19). Mark Thoma has more on the topic, including an excerpt from an interview with Eric Maskin in which a related paper by Fostel and Geanakoplos is discussed. This is one of five contributions recommended by Maskin, all of which are worth reading.

2 comments:

The 'higher asset price due to leverage' seems to require a key behavioural idiosyncrasy - those pessimistic about the mortgages will nevertheless lend money to those optimistic about them to make the exact same purchase that they would have avoided. I understand that in a more complex model with multiple assets, the argument will not be as stark, but the crux of the logic will remain. While this may indeed happen to an extent, it is doubtful if this will be the key driver of asset prices in a bubble all by itself.

Of course, securitization, model errors resulting from underestimating correlation (and hence overestimating the benefits of diversification)and rating arbitrage make it a more significant effect than it otherwise would have been.

They lend money that is secured by the collateral. Even if they believe that the collateral value will decline, it makes sense for them to do so as long as they don't lend too much - that is, if the margin is sufficiently high. Similarly, a broker would happily lend to an investor even if he believes that the resulting purchase is misguided, as long as the loan is small enough relative to the value of shares purchased. I don't really see the problem here.